Insurance Companies and Pension Plans

[tkvfrm]

Hi,
I have this question from GARP Reading 2017 Book 3 --> Page 25 --> Example 2.1

Interest Rates for all maturities are 4% semi annual compounding
Premiums paid once a year at the begining

What is the insurance companies breakeven premium for term life insurance USD100,000 for a man aged 90.
Probability of death is 0.168352
The answers are in 2 parts, the first one is term insurance lasts one year, the second lasts 2 years.
My question is on the second part.
In case of death in the first year the payout is $16505.00 (discounted at 2%) . The solution further calculates the present value assuming the policy survives the first year and dies in the second year. Assuming this occurs in 18 months the payout of $15426 discounted for 3 periods is at $14536. Till here it is fine.
However the value of payouts is shown as a sum of 1st year payout ($16505 ) and 2nd year payout ($14536) = $31041.
Question. Why is the payout summed up for 1st and 2nd year? do the payouts get added each year of survival?

 

[David Harper CFA FRM]

Hi @tkvfrm I think that's a nice observation, because it teases out the different probabilities employed (conditional versus unconditional/joint). To recap the given assumptions, the table tells us:

• A 90 year old male has conditional death probabilities of 16.8352% (in his 90th year) and 18.5486% (in his 91st year).

The unconditional death probability is the joint probability of survival in the 90th year followed by (i.e, "and" as in joint) death in the 91st year, and this is (1-16.8352%)*18.5486% = 15.4259%; this is not a conditional probability, this is the joint probability of both survival in year 90 and death in year 91. In this way, it is a different outcome (a mutually exclusive outcome) and, as of the beginning of year 90, there are three possible outcomes in the total (100%) probability space over the next two years:

1. he dies in his 90th year, with the given (conditional) probability = 16.8352%
2. he survives his 90th year, but dies in his 91st year, with joint probability = 15.4259%
3. he survives both years, with probability = (1-16.8352%)*(1-18.5486%) = 67.74%. These three outcomes, from the perspective of the start of his 90th year, sum to 100.0% as they must.

So the payments are not really being double-counted, rather this is a weighted average expectation given by $100,000 * (16.8352% + 15.4259%) + (zero*67.74%). The probability of a payout is (16.8352% + 15.4259%) = 32.3% because he can die on either year. Hence the importance of unconditional probabilities; they are from the initial perspectives and can be summed (the 16.8352% conditional probability in the first year is also an unconditional probability). I hope that's helpful!

 

[tkvfrm]

Hi David,
Excellent and convincing analysis. I completely missed the conditional and joint probabilities part. The chart values diverted me from the line of thinking that was outlined in your response. Thank you.